I cut out a new bed and will be planting and mulching in the front of my house. The tree in the picture I would like to trim near the trunk to open up the bed a little bit so I can mulch easy underneath and improve the appearance of the tree overall.

Is it okay to take off one rung of branches going around this guy this time of year? based on how low they are now I don't think it's ever been trimmed. This tree is on the NW corner of my house FYI.